"We are going to Moscow by train."
Translation:Мы едем в Москву на поезде.
Also should be possible: "мы едем в Москву поездом". In general, also: "мы летим самолетом", "плывем пароходом"
You can rarely hear such forms nowadays. People really use "на + Loc." a lot more often to describe means of transport.
Not really, what vadimfowler says is actually quite commonplace. I wouldn't call it archaic.
Well, yeah, it's not really archaic, but the alternative "на + Loc." is more common, especially in the everyday speech.
Russia is a big country, so the frequency of usage may depend on where you are. I concede though that I haven't lived there for some years :)
Why can't you use мы ездим because the action is present tense? Any help would be greatly appreciated
"мы ездим" would imply a repetitive action. These are two different verbs: "ездить" and "ехать".
You may use it. It is just not that common (though, the form is popular in Ukrainian and in Russian spoken in Ukraine). I have never used that form with transportation but, as Nievva mentioned in the topic, there are places where you can still hear it expressed like that.